NEW DEVELOPMENTS IN ONSET THEORY FOR ONSET OF RESIN FAILURE IN FIBRE REINFORCED COMPOSITES

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1 THE 19 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS NEW DEVELOPMENTS IN ONSET THEORY FOR ONSET OF RESIN FAILURE IN FIBRE REINFORCED COMPOSITES S. H. Lm 1, G. Pearce 1, D. Kelly 1, B. G. Prusty 1 *, A. Crosky 1 School of Mechancal and Manufacturng Engneerng, Unversty of New South Wales, Sydney, Australa, School of Materals Scence and Engneerng, Unversty of New South Wales, Sydney, Australa * Correspondng author (g.prusty@unsw.edu.au) Keywords: Onset Theory, Fbre Renforced Polymers, Mcromechancal Modellng 1 Introducton Ths paper presents recent work on the Stran Invarant Falure Theory (SIFT) that has recently been named Onset Theory to emphasse ts applcaton to onset of rreversble damage n carbon fbre and glassy polymerc resns. The theory was frst proposed by Gosse and Chrstensen [1]. It s based on two fundamental assertons; that predcton of falure of homogeneous materals should be based on stran, and that the onset of rreversble behavour n homogeneous solds arses from only two possble modes. In the case of glassy polymer resns, falure can occur va excessve dlataton (volume change) or excessve dstorton (shape change). Only two ntrnsc materal propertes are requred to characterse these falures, based on the frst stran nvarant (dlataton) and the second devatorc stran nvarant. In the approach defned by Ppes and Gosse [], lamnate strans are dehomogensed to gve representatve strans n the composte consttuents usng enhancement factors derved from the mcromechancal analyss of fbre-matrx unt cells. Crtcal values are defned for the dlatatonal and dstortonal nvarants for the resn usng 90 degree and 10 degree undrectonal off-axs test specmens respectvely. Crtcally, valdaton of the theory has been provded by Sul et al. [3] by reproducng the falure modes n molecular dynamcs smulatons completely ndependently of expermentally derved approach of Ppes and Gosse. Recent developments nclude refnements of the testng procedures that are used to defne the crtcal nvarants and the use of submodellng to dehomogense the strans. The research has shown that the shear-bendng couplng that occurs n the off-axs specmens can be allevated by usng hgh aspect rato 14:1 specmens and bondng tabs to the specmens so that falures ntate outsde the grps. Undrectonal specmens are used because the onset of damage and fnal falure occur at the same test stran. The nvarants are then sampled on the actual falure surface n the fnte element analyss used to back out the crtcal nvarants. In the recent work [4-6] the use of enhancement factors (prevously stran amplfcaton factors) to dehomogense the strans s valdated by a mcromechancal submodellng approach. Ths method embedded mcromechancal models wthn the smulaton doman such that crtcal strans n the consttuents could be drectly extracted from one smulaton. The beneft of the drect mcromechancal approach s most apparent for prelmnary studes whch may cover a varety of materal optons and/or senstvty and parameter studes. In these cases the drect approach s more computatonally effcent as t removes the computatonal repetton requred to construct the enhancement factor matrx [7]. In ths paper the applcaton of Onset Theory to specmens manufactured from plan weave fabrc ples wll be descrbed. The approach nvolves two levels of dehomogensaton from the lamnate to ndvdual tows n the fabrc ples, and then from the tows to strans n the fbre and resn at the fbre level. The development of an approach to characterse the growth of the damage nto mcrocracks and falure at the lamnate level wll be brefly ntroduced.

2 Revew of the Onset Theory.1 Stran Invarants Onset Theory s a recent development for glassy polymers constraned by fbres n structural compostes. The theory smply posts that the constraned polymer wll fal when subjected to excessve deformaton; ether through excessve volume change (dlataton) or excessve shape change (dstorton). Ths fundamental property of polymer materals s based heavly n thermodynamc theory and has been confrmed by atomstc smulaton of polymers [3]. To quantfy dlataton and dstorton, stran nvarants are used, related to the fundamental nvarants of the stran tensor. The dlatatonal nvarant s equal to the frst stran nvarant, J 1, and s shown n Equaton 1. The dstortonal nvarant s equal to the square root of the second devatorc stran nvarant, J, and s shown n Equaton. dl ds where: dl ds 1,,3 x, y, z xy, yz, zx x y z x y xy 1 1 y z yz 6 4 z x zx Dlatatonal stran nvarant Dstortonal stran nvarant Prncpal elastc strans Engneerng elastc stran tensor n Cartesan coordnates (1) () Irreversble behavour s predcted to occur when ether the dlatatonal or dstortonal stran nvarant at any locaton n the resn exceeds the correspondng crtcal value, dl,crt or ds,crt. The two crtcal stran nvarants are consdered ntrnsc materal propertes.. Effect of Temperature Onset theory explctly accounts for the thermal resdual strans present n constraned resn by only evaluatng stran nvarants usng elastc strans; those whch are compatble wth the stress tensor through Hooke s Law, as shown n Equaton 3. S (3) j j For a fully constraned 3D element of resn subjected to a postve temperature ncrement, T, the prncpal elastc strans are gven by Equaton 4. where: 1,,3 T Lnear coeffcent of thermal (4) expanson (CTE) for the resn Therefore, stran due to the thermal shrnkage of free resn does not lead to falure. On the other hand, shrnkage of constraned resn, such as that constraned by fbres, leads to a non-zero elastc stran tensor, whch can lead to falure. All stran tensor terms shown n ths paper are taken from the elastc stran tensor..3 Dehomogensaton An mportant feature of the onset theory s the dehomogensaton process of the composte structure. Ths allows determnaton of the components of the stran tensor feld wthn the matrx phase and fbre phase [] and enables damage predcton wthn a complex structural doman, such as the random dstrbuton of fbres and resn shown n Fg. 1. Wth ths process, the thermal resdual effect due to the thermal contracton msmatch of fbre and resn s also taken nto account. Two Representatve Volume Elements (RVEs) are used to account for the dstrbuton of fbres and resn wthn the composte, shown n Fg. 1. The justfcaton and valdaton of ths approach s gven n [7].

3 NEW DEVELOPMENTS IN ONSET THEORY ON ONSET FOR RESIN FAILURE IN FIBRE REINFORCED COMPOSITES where: k k M T k j j j T ε M ε ε A (5) Fg. 1. Realstc dstrbuton of fbres and resn wthn a composte and Idealsed Representatve Volume Elements (RVEs) wth the same volume fracton (V f ). A seres of crtcal ponts, k, are used as samplng ponts wthn the RVE. At each pont a matrx of enhancement factors, M, s generated whch relates j the nternal stran tensor term,, to the externally appled unt stran tensor term, j. A thermal stran vector, A, relates the nternal stran tensor term,, to a unt temperature dfferental appled to the model. Algorthmcally, the dehomogensaton process can be expressed for lamnates made from undrectonal ples as follows: 1. Analyse a contnuum model wth mechancal load to defne the elastc strans, M, n the ples.. Analyse the contnuum model wth free boundares but ΔT= T-T cure to defne the elastc strans, T, n the ples due to the cure process. 3. Supermpose the elastc strans, apply the stran enhancement factors and add the cure strans due to the fbre and resn msmatch, as shown n Equaton 5. ε M A k k j k T Mcro-elastc stran tensor at the k pont n the RVE Enhancement matrx for the k pont n the RVE Thermal stran vector due to CTE msmatch between the fbre and th resn at the k p th ont n the RVE Thermal dfference between servce temperature and cure temperature Wth the ad of mcromechancal enhancement, a detaled assessment of rreversble deformaton s possble beyond the homogenous deformaton state [, 7]. Damage onset wthn the matrx phase s predcted when Equaton 6 s satsfed. dl ds max, 1 dl, crt ds, crt th (6) Damage onset wthn the fbre phase s not addressed n ths paper. 3 Evaluaton of Crtcal Stran Invarants for Resn Falure As was summarsed n the prevous secton, the onset of the matrx/resn falure can be conducted when the crtcal dlatatonal and dstortonal stran nvarants are known. As the deformaton of the matrx n real undrectonal lamnates s constraned by the fbres, the deal method s to acheve the constraned stran state n a neat resn test and extract the crtcal stran nvarants. However, ths s a dffcult problem and s stll under nvestgaton. An alternatve approach has been adopted to extract the crtcal nvarants for the resn falure that was by usng undrectonal composte tensle tests and then ther correspondng fnte element analyss. A set of dfferent off-axs undrectonal tests from 10 to 90 to generate crtcal values of stran nvarants for the matrx has been proposed by Ppes and Gosse []. The 0 specmen was omtted as t wll fal n the fbres. Based on these tests and normalsed dstortonal and dlatatonal nvarants, 10 specmen 3

4 provdes the crtcal nvarant for dstortonal behavour whle the 90 specmen determnes the crtcal nvarant for dlatatonal deformaton. Ths approach was successfully appled by Tran et al. [6] on a dfferent materal and was able to determne the onset of resn falure of the materal. In ths paper, the crtcal nvarants requred for the fabrc archtectures and damage progresson n the subsequent sectons was based on T800s/3900- specmens. The dmensons of the specmens are 5 mm length, mm wdth and 5.8 mm thckness. In ths evaluaton, only 10 and 90 specmens were utlsed for the off-axs unaxal tensle tests. A MTS810 unaxal hydraulc testng machne wth capacty of 100 kn was utlsed wth surfalloy wedges to reduce specmen falure n the grps secton. Materal propertes of T800s/3900- are gven n Table 1 and were appled n the fnte element analyss to determne the crtcal stran nvarants. Table 1. Materal propertes of T800s fbre and resn wth fbre volume fracton=60%. Propertes T800s E 1 (GPa) E (GPa) E 3 (GPa) G 1 (GPa) G 3 (GPa) G 13 (GPa) ν ν ν α 11 (/ C) x10-6 α = α 33 (/ C) 8.3 x x10-6 The crtcal dlatatonal and dstortonal stran nvarants were found to be 500 and respectvely. 4 Fabrc Archtectures 4.1 Fabrc Modellng The new work n ths paper ncludes an extenson of onset theory to fabrc archtectures. Contnuum lamnate strans are dehomogensed to defne local strans n the tow archtecture of the fabrc. Ths wll be acheved usng meso-mechancal RVE cell analyss n whch the resn and tows are modelled as contnua, as shown n Fg.. In ths paper, the materals appled for the tows and resn are T300/3900- lamna and epoxy respectvely and ther correspondng materal propertes are gven n Table. (c) Fg.. RVE cell, tow and (c) resn part of the meso-mechancal unt cell model for stran dehomogensaton.

5 NEW DEVELOPMENTS IN ONSET THEORY ON ONSET FOR RESIN FAILURE IN FIBRE REINFORCED COMPOSITES Table. Materal propertes of T300/3900- lamna for tows and for resn. Propertes T300/3900- lamna Epoxy resn E 1 (GPa) E (GPa) E 3 (GPa) G 1 (GPa) 5 1. G 3 (GPa) G 13 (GPa) 5 1. ν ν ν α 11 (/ C) 1.05X X10-6 α = α 33 (/ C) 3X X10-6 The materal propertes were appled along the local axs of the tows to retan the accurate orthotropc behavour durng the cell analyss. The volume fracton of the tows to the unt cell s 55.6%. The dmensons of the unt cell are mm length, mm wdth and 0.5 mm heght to represent one layer thckness. The tow s major and mnor axs rato s set to be 1:1, where the mnor axs dameter s 0.1 mm. In the expermental tensle test conducted n ths project for ths materal, the falure stran was 1.3%. Ths stran was appled perpendcular to the surface perpendcular to x-axs. The other surfaces were constraned to reman planar wth no net normal reacton force to represent symmetrc free contracton boundares. Fg. 3 and Fg. 4 show the contour plots of dlatatonal and dstortonal nvarants of the unt cell model respectvely, where these results were represented usng the separate tows and resn enttes. The maxmum dlatatonal and dstortonal nvarants were located along the slender edge of the transverse tows. The maxmum dlatatonal nvarant was 6488 με and occurred drectly adjacent to the large resn volume n the centre of the model. Meanwhle, the maxmum dstortonal nvarant was 5854 με and was located n the regon where the tows overlapped each other. In the tows, the maxmum dlatatonal nvarant was near the centre regon of the RVE cell whle the maxmum dstortonal nvarant was located wthn the bulk of the transverse tows n the overlappng regon. These crtcal locatons n the RVE cell were dehomogensed and the maxmum dlatatonal nvarant was found to be με and the maxmum dstortonal nvarant was με. In ths case falure s predcted to occur through dlataton wthn the transverse tow volume. Ths agrees wth exstng evdence for the locaton of mcro-crackng n fabrcs however a drect comparson predcted falure stran has not been attempted. Fg. 3. Dlatatonal nvarant of the RVE cell model wth tows and resn enttes. Fg. 4. Dstortonal nvarant of the RVE cell model wth tows and resn enttes. 5

6 4. Fabrc Dehomogensaton Process A b-level fabrc dehomogensaton process s proposed for fabrc modellng. The contnuum strans wll be dehomogensed usng fabrc RVE models wth contnuum tow materal. The tow materal wll be further dehomogensed usng RVE models of UD materal. The lnk between the length scales can be acheved wth embedded submodels however ths s computatonally very neffcent. The methodology proposed here s to extend the Onset Theory approach to UD materals, whch uses enhancement factors at chosen crtcal locatons. The new formulaton for the stran wthn the tow wll be gven by Equaton 6. ε kl M k M l ε A l A k (7) j j j T T where all symbols have ther prevous meanng wth k representng a selected crtcal samplng pont wthn the UD RVE model and l representng a selected crtcal pont wthn the tows of the fabrc RVE model. Ths formulaton s under development and has not yet been tested. A key challenge s the selecton of crtcal locatons n the fabrc unt cell whch wll capture the majorty of crtcal stran cases for an arbtrary contnuum stran, ε j. 5 Damage Progresson A damage progresson algorthm based on Onset Theory and an element elmnaton method has been developed by Tay et al [8]. A smlar algorthm s beng developed here usng a sub-model embedded n a global analyss as shown n Fg. 6. The submodel s a mcroscopc three-dmensonal fbre-resn model that attempts to follow the growth of damage followng ntaton at the mcro-mechancal level predcted by the stran nvarants. Dsplacements from the global analyss are appled to the boundares of the sub-model. Fg. 5. Two level dehomogensaton requred to apply onset theory to fabrc parts Fg. 6 Three-dmensonal submodel for damage progresson.

7 NEW DEVELOPMENTS IN ONSET THEORY ON ONSET FOR RESIN FAILURE IN FIBRE REINFORCED COMPOSITES The algorthm evaluates the dlatatonal and dstortonal nvarants at the centre of each element. The propertes n elements that have exceeded the crtcal values are reduced before returnng to repeat the analyss. If the damage coalesces nto a mcrocrack the growth can be contaned by the fbre archtecture. The algorthm also evaluates the stran energy release rate n order to provde a measure that can be used to lmt the growth of cracks that are not constraned by the fbres. The algorthm has been appled to follow the hstory of damage developng n a specmen manufacture from T800/3900- undrectonal tape wth the layup 0 /90 4 /0. Ths problem has been nvestgated by several authors, the paper by Narn [9] beng typcal of the work that has been done. The analyss predcts that under axal load transverse cracks wll develop n the 90 layers that wll exhbt a characterstc spacng. The axal stran shown n Fg. 7 s releved by the presence of the crack and there s a gap before the stran s restored to precrack levels by the requrement for compatblty wth the 0 layer. Expermental results ndcate that cracks frst appear n the 90 ples at an average axal stran of At hgher strans ntermedate cracks appear untl the spacng reduces to an average spacng of about 1 to 1.5 tmes the thckness of the central 90 tow regon as shown n Fg. 7. Applcaton of the new algorthm showed no ntaton of mcrocracks (the dlatatonal nvarant was everywhere less than the crtcal value) for an average axal stran of The frst crack appeared for an axal stran of The crack growth extended across the wdth of the 90 ples and arrested (the dlatatonal nvarant fell below the crtcal value) on the boundary wth the 0 ples. The submodel was then moved so that the ntal crack was on the left-hand boundary. A second crack ntated at 6000, extended across the 90 ples and agan arrested. A thrd crack formed md-way between the frst two cracks when the average axal stran was rased to above No further crack ntaton occurred even though the axal stran was rased to the average stran for two-pece falure of the specmen. The Scannng Electron Mcroscopy (SEM) mage n Fg. 8 also reveals that the cracks grow on the nterface wth the fbre and fnger shaped mcroscopc cavtatons can grow n the resn perpendcular to the fbre axs. These fnger cracks then coalesce nto the prmary fracture surface extendng parallel to the fbre axs. The new algorthm s beng appled to a refned model of fbre and resn to nvestgate ths ntal phase of crack growth. Intal results ndcate the damage grows on the surface of the fbre parallel to the fnal fracture surface before brdgng the gap between the fbres across a brdge that could be nterpreted as leadng to a fnger crack. Fg 7. Axal stran dstrbuton leadng to cracks and equally spaced transverse cracks n 0 /90 4 /0 T800/3900- tape specmen. Fg. 8. Damage growth at a mcroscopc level. SEM mage of fnger cracks near surface of a T300/Cycom970 specmen. 7

8 6 Concluson Onset Theory for the predcton of the onset of falure n undrectonal composte lamnates has been well establshed. Ths paper extends the applcaton of Onset Theory to the predcton of damage ntaton stes n composte fabrcs and also the mcro/macroscopc progresson of damage wthn a heterogeneous fbre-resn system. The applcaton to lamnates manufactured from fabrc ples requred two levels of dehomgensaton. The frst from the lamnate to the fabrc tow wll depend on the archtecture of the fabrc wth dfferent unt cell models requred for plane weave and satn weave, for example. The second level of dehomogensaton to the fbre/resn level requres the propertes of the fbre and resn and ncludes the fbre volume fracton of the fbre n the fabrc tow. Future work wll nvolve an expermental nvestgaton of damage development n fabrc specmens that wll permt the locaton and stran levels, at whch the mcrocracks develop, to be compared wth the theoretcal predctons. The work on damage progresson s new and the current methodology s ntended to predct the ntal growth of the damage. A methodology wll need to be developed to create a crack n the global model followng the development of damage at the mcroscopc level to allow the smulaton to be extended to the lamnate level. Acknowledgement Ths project has been supported by the Australan Research Councl Lnkage Grant LP Dr Jon Gosse and Mr Steve Chrstensen from The Boeng Company are gratefully acknowledged for many useful dscussons and techncal nput. References [1] Gosse, J.H., Chrstensen, S. (001) Stran Invarant Falure Crtera for Polymers n Composte Materals. 4nd AIAA Structures, Structural Dynamcs and Materals Conference and Exhbt, Aprl 001, 001, Seattle. [] Ppes, R.B., Gosse, J.H. (009) An onset theory for rreversble deformaton n composte materals. ICCM-17, the 17th Internatonal Conference on Composte Materals, Ednburgh, UK, 7-31 Jul 009. [3] Sul, J., Prusty, B.G., Chrstensen, S., Wggns, J. (013) Falure study for thermosettng epoxy n composte materals usng onset theory n conjuncton wth molecular dynamcs. ICCS17-17th Internatonal Conference on Composte Structures, 17-1 June, 013, Porto, Portugal. [4] Tran, T. Development of mcromechancal modellng procedures usng the onset theory for falure of compostes, n School of Mechancal and Manufacturng Engneerng. Unversty of New South Wales: Sydney, Australa. 01. [5] Tran, T.D., Kelly, D., Prusty, B.G., Gosse, J.H., Chrstensen, S. (01) Mcromechancal modellng for onset of dstortonal matrx damage of fber renforced composte materals. Compos. Struct., 94(), [6] Tran, T., Kelly, D., Prusty, B.G., Pearce, G., Gosse, J. (013) A mcromechancal submodellng technque for mplementng Onset Theory. Compos. Struct. In Press, Corrected Proof. DOI: [7] Buchanan, D.L., Gosse, J.H., Wollschlager, J.A., Rtchey, A., Byron Ppes, R. (009) Mcromechancal enhancement of the macroscopc stran state for advanced composte materals. Compos. Sc. Technol., 69(11-1), [8] Tay, T.E., Tan, S.H.N., Tan, V.B.C., Gosse, J.H. (005) Damage progresson by the elementfalure method (EFM) and stran nvarant falure theory (SIFT). Compos. Sc. Technol., 65(6), [9] Narn, J.A. (1989) The Stran Energy Release Rate of Composte Mcrocrackng: A Varatonal Approach. J. Compos. Mater., 3(11),